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Express (274.1625)10 in the IEEE single and double precision standard of floating point representation.
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Express (127.125)10 in the IEEE single precision standard of floating point representation:

Step 1: Convert the decimal number to its binary fractional form.

Convert this format is in base2 format For this, first convert 127 in to binary format

127=1111111

Convert 0.125 in to binary format

0.125*2=0.250 0

0.250*2=0.500 0

0.500*2=1.00 1

Binary format of 127.125=1111111.100

Shifting this binary number

1.1111111002 *6 Normalized

1.111111100 is mantissa

2 **6 is exponent

add exponent 127 +6=133

Step 2: Normalize the binary fractional number.

Move the decimal point left or right so that only a single binary

digit "1" is to the left of the binary decimal point. Compensate by

adjusting the exponent in the opposite direction.

1.111111100 times 2*6

Moving the decimal left seven decreases the size of the number; so,

we use an exponent of 6 to compensate and keep the number the same size.

Step 3: Convert the exponent to 8-bit excess-127 notation

Add 127 to the exponent and convert it to 8-bit binary:

6+ 127 = 133--> 10000101

Step 4: Convert the mantissa/significand to "hidden bit" format.

Since every binary floating-point number (except zero!) is normalized

with "1." at the start, there is no need to store that leftmost "1".

Remove the leading "1." from the mantissa/significand:

1.111111100--> 111111100

Step 5: Write down the 1+8+23 = 32 bits.

127.125 is positive - the sign bit is zero: 0

The next eight bits are the exponent: 10000101

The next 23 bits are the mantissa: 11111110000000000000000

Binary result (32 bits): 01000010111111110000000000000000

Express (127.125)10 in the IEEE double precision standard of floating point representation.

Step 1: Convert the decimal number to its binary fractional form.

Convert this format is in base2 format For this, first convert 127 in to binary format

127=1111111

Convert 0.125 in to binary format

0.125*2=0.250 0

0.250*2=0.500 0

0.500*2=1.00 1

Binary format of 127.125=1111111.100

Shifting this binary number

1.1111111002 *6 Normalized

1.111111100 is mantissa

2 **6 is exponent

add exponent 1023+6=1029

Step 2: Normalize the binary fractional number.

Move the decimal point left or right so that only a single binary

digit "1" is to the left of the binary decimal point. Compensate by

adjusting the exponent in the opposite direction.

1.111111100 times 2*6

Moving the decimal left seven decreases the size of the number; so,

we use an exponent of 6 to compensate and keep the number the same size.

Step 3: Convert the exponent to 8-bit excess-127 notation

Add 1023 to the exponent and convert it to 8-bit binary:

6+1023= 1029--> 10000000101

Step 4: Convert the mantissa/significand to "hidden bit" format.

Since every binary floating-point number (except zero!) is normalized

with "1." at the start, there is no need to store that leftmost "1".

Remove the leading "1." from the mantissa/significand:

1.111111100--> 111111100

Step 5: Write down the 1+11+52 = 64bits.

127.125 is positive - the sign bit is zero: 0

The next 11 bits are the exponent: 111111100

The next 23 bits are the mantissa: 111111100000000000000000000000000000000000000000000000

Binary result (64 bits): 0111111100 111111100000000000000000000000000000000000000000000000

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